Consider a crown with a chief factor \(A\)  of order \(n\)  and let \(B\)  be another chief factor in the same crown as \(A\) . We have that \(A\)  and \(B\)  are \(G\) -connected (see [1]}), but not \(G\) -isomorphic. According to [2]}, there exists a 1-cocycle \(\beta \in \operatorname{Z}^1(G,B)\)  such that \(A\)  is isomorphic to \(B_\beta \)  as \(G\) -groups, that is, there exists an isomorphism \(\varphi \colon A\longrightarrow B\)  such that \(a^{g\varphi }=a^{\varphi g g^\beta }\)  for each \(a\in A\)  and \(g\in G\) . Hence the number of chief factors \(G\) -connected to \(B\)  but not \(G\) -isomorphic to \(B\)  is bounded by \(\vert \operatorname{Z}^1(G, B)\vert \)  and so \(\operatorname{rm}_n(G)\le \vert \operatorname{Z}^1(G,B)\vert \) . Since these 1-cocycles are uniquely determined by the image of a generating system of \(G\) , we have that the number of cocycles is at most \(n^d\) . It follows that \(\operatorname{rm}_n(G)\le n^d\) .
